Volume Calculation Using Disc and Washer Methods

This video tutorial explains how to calculate the volume of a solid formed by rotating a region around the x-axis or y-axis using the disc and washer methods. It covers the basic principles of these methods, including the use of cross-sectional areas and integration. The tutorial provides examples using functions like y = sqrt(x) and y = 1/x, demonstrating the process of setting up and solving integrals to find the volume of the resulting solids. Additionally, it discusses how to handle rotations about the y-axis, providing further examples and explanations.

What is the primary purpose of using the disc method in volume calculation?

• To measure the weight of a solid

• To find the surface area of a solid

• To calculate the volume of a solid by rotating a region

• To determine the length of a curve

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In the disc method, what shape is formed by the cross-sectional area?

• Square

• Triangle

• Rectangle

• Circle

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When using the disc method, what is the formula for the volume of a cylinder?

• πr²h

• 2πrh

• πd²/4

• πr²

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What is the integral used to find the volume of a solid rotated about the x-axis using the disc method?

• ∫[a to b] 2πr dy

• ∫[a to b] πr² dy

• ∫[a to b] 2πr dx

• ∫[a to b] πr² dx

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In the example of rotating y = √x about the x-axis, what are the limits of integration?

• 0 to 2

• 0 to 4

• 1 to 3

• 2 to 5

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What is the final volume when rotating y = √x about the x-axis from x = 0 to x = 4?

• 8π

• 4π

• 2π

• 16π

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When rotating y = 1/x about the x-axis, what is the expression for r(x)?

• x

• 1/x

• √x

• x²

What is the volume of the solid formed by rotating y = 1/x about the x-axis from x = 1 to x = 3?

• π/3

• 2π/3

• π

• 4π/3

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In the example of rotating y = x² about the y-axis, what is the expression for r(y) in terms of y?

• 1/y

• y²

• y

• √y

What is the final volume when rotating y = x^(2/3) about the y-axis from y = 0 to y = 1?

• 2π

• π

• π/4

• π/2